Balanced arrays of strength 4 and balanced fractional 3m factorial designs

A connection between a balanced fractional 2^m factorial design of resolution 2l + 1 and a balanced array of strength 2l with index set {μ₀, μ₁,…, μ_2l} was established by Yamamoto, Shirakura and Kuwada (1975). The main purpose of this paper is to give a connection between a balanced fractional 3^m factorial design of resolution V and a balanced array of strength 4, size N, m constraints, 3 levels and index set {λ_l0l1l2}.     

On computer generation of balanced arrays

This paper provides an algebraic (and hence computing) procedure for generation of balanced arrays having two symbols, m rows, specified minimum and maximum column weights, arbitrary strength t≦m, and index set parameters μ^t₁, μ^t₂,…, μ^t_t. μ^t₀ is unspecified, and calculated as part of the algorithm, although the procedure for specifying it is straightforward and can be used if desired. Array generation is herein reduced to finding integral solutions to a linear programming problem. It is shown that the integral solutions of the system of equations comprise all balanced arrays with the given set of parameters.A computing algorithm is provided which constructs the system of equations to be solved; it has been interfaced with a standard linear programming package to provide some preliminary results.Additional algorithms whose development should result in substantial decreases in computing costs are discussed.     

On orthogonal arrays attaining Rao's bounds

Rao (1947) provided two inequalities on parameters of an orthogonal array OA(N,m,s,t). An orthogonal array attaining these Rao bounds is said to be complete. Noda (1979) characterized complete orthogonal arrays of t=4 (strength). We here investigate complete orthogonal arrays with s=2 (levels) and general t; and with t=2, 3 and general s.     

On the construction of orthogonal F-squares of order n from an orthogonal array (n, k, s, 2) and an OL(s, t) set

Complete sets of orthogonal F-squares of order n = s^p, where g is a prime or prime power and p is a positive integer have been constructed by Hedayat, Raghavarao, and Seiden (1975). Federer (1977) has constructed complete sets of orthogonal F-squares of order n = 4t, where t is a positive integer. We give a general procedure for constructing orthogonal F-squares of order n from an orthogonal array (n, k, s, 2) and an OL(s, t) set, where n is not necessarily a prime or prime power. In particular, we show how to construct sets of orthogonal F-squares of order n = 2s^p, where s is a prime or prime power and p is a positive integer. These sets are shown to be near complete and approach complete sets as s and/or p become large. We have also shown how to construct orthogonal arrays by these methods. In addition, the best upper bound on the number t of orthogonal F(n, λ₁), F(n, λ₂), …, F(n, λ₁) squares is given.     

Regular automorphism groups on partial geometries

Recently, Ghinelli (Geom Dedicata (1992) 165–174) had studied the generalized quadrangles which admits automorphism groups acting regularly on the points. In this paper, we generalize her idea to partial geometries, pg(s + 1, t + 1, α). Some examples and basic properties are given. In particular, we prove that under certain conditions on the automorphism group and the lines, such a geometry is a translation net. Applying the results to the case when s = t and the automorphism group G is abelian, we find that either the geometry is a translation net or all the lines of the geometry are generated by a subset of G. Also, for this case, we conjecture that the parameter α is either s or s + 1, except (s, α) = (5, 2), and we have checked that it is true for s ⩽ 500.     

Improved estimation of the scale parameter, the hazard rate parameter and the ratio of the scale parameters in exponential distributions: An integrated approach

We give new classes of Strawderman-type improved estimators for the scale parameter σ₂ and the hazard rate parameter 1/σ₁ of the exponential distributions E(μ₂,σ₂) and E(μ₁,σ₁) under the entropy loss. We then use these estimators to construct improved estimators for the ratio ρ=σ₂/σ₁. The sampling framework we consider integrates important life-testing schemes separately studied in the literature so far, namely, (i) i.i.d. sampling, (ii) Type-II censoring, (iii) progressive Type-II censoring and adaptive progressive Type-II censoring and (iv) record values data. Furthermore, we establish simple identities connecting the risk functions of the estimators of σ₂ and 1/σ₁ and those of ρ that have a direct impact on studying the risk behavior of the latter estimators. Finally, we indicate that no matter which of the above life-testing schemes is employed for the estimation of σ₂, 1/σ₁ or ρ, the corresponding improved estimator, which may be of Stein-type or Brewster and Zidek-type or Strawderman-type, will offer the same improvement over the usual estimator as long as the number of observed complete failure times is the same for each scheme. Our results unify and extend existing results on the estimation of exponential scale parameters in one or two populations.     

On a class of partially balanced incomplete block designs

Bose and Clatworthy (1955) showed that the parameters of a two-class balanced incomplete block design with λ1=1,λ2=0 and satisfying r <k can be expressed in terms of just three parameters r,k,t. Later Bose (1963) showed that such a design is a partial geometry (r,k,t). Bose, Shrikhande and Singhi (1976) have defined partial geometric designs (r,k,t,c), which reduce to partial geometries when c=0. In this note we prove that any two class partially balanced (PBIB) design with r <k, is a partial geometric design for suitably chosen r,k,t,c and express the parameters of the PBIB design in terms of r,k,t,c and λ2. We also show that such PBIB designs belong to the class of special partially balanced designs (SPBIB) studied by Bridges and Shrikhande (1974).     

Robustness of locally most powerful invariant test for normal mixture model in control and treatment populations

Two independent samples from control with N(μ₁, σ²) and treatment with pN(μ₁, σ²) + (1 − p)N(μ₂, σ²) are considered. A locally most powerful invariant test for testing H₀: μ₁ = μ₂ against H₁ : μ₂ > μ₁, where σ² > 0, 0 < p < 1 are unknown, is obtained. Also, the robustness of the test statistic on the lines of Kariya and Sinha (Robustness of Statistical Tests (1989). Academic Press, New York) is studied.     

A note on the minimum size of an orthogonal array

It is an elementary fact that the size of an orthogonal array of strength t on k factors must be a multiple of a certain number, say L_t, that depends on the orders of the factors. Thus L_t is a lower bound on the size of arrays of strength t on those factors, and is no larger than L_k, the size of the complete factorial design. We investigate the relationship between the numbers L_t, and two questions in particular: For what t is L_t < L_k? And when L_t = L_k, is the complete factorial design the only array of that size and strength t? Arrays are assumed to be mixed-level.

We refer to an array of size less than L_k as a proper fraction. Guided by our main result, we construct a variety of mixed-level proper fractions of strength k ? 1 that also satisfy a certain group-theoretic condition.     

On some properties of the dual of a totally balanced block design

In this paper it is shown that the dual of a totally balanced block design with t = b, is also a totally balanced block design. It is shown that Fisher's inequality $\text{b≧t}$ for BIB designs, holds also for a totally balanced block design.     

Estimated Generalized Least Squares for a Heteroscedastic Regression Model

This paper considers the general linear regression model yc = X₁β+u_t under the heteroscedastic structure E(u_t) = 0, E(u²) =σ²- (X_tβ)², E(u_t u_s) = 0, tæs, t, s= 1, T. It is shown that any estimated GLS estimator for β is asymptotically equivalent to the GLS estimator under some regularity conditions. A three-step GLS estimator, which calls upon the assumption E(u_t²) =s?²(X,β)² for the estimation of the disturbance covariance matrix, is considered.     

Tabu search for covering arrays using permutation vectors

A covering array  CA(N;t,k,v)$\mathsf{CA}(N;t,k,v)$ is an N×k$N\times k$ array, in which in every N×t$N\times t$ subarray, each of the v^t$v^t$ possible t  -tuples over v$v$ symbols occurs at least once. The parameter t is the strength   of the array. Covering arrays have a wide range of applications for experimental screening designs, particularly for software interaction testing. A compact representation of certain covering arrays employs “permutation vectors” to encode v^t×1$v^t\times 1$ subarrays of the covering array so that a covering perfect hash family whose entries correspond to permutation vectors yields a covering array. We introduce a method for effective search for covering arrays of this type using tabu search. Using this technique, improved covering arrays of strength 3, 4 and 5 have been found, as well as the first arrays of strength 6 and 7 found by computational search.     

Comparisons of search designs using search probabilities

Search designs are considered for searching and estimating one nonzero interaction from the two and three factor interactions under the search linear model. We compare three 12-run search designs D1, D2, and D3, and three 11-run search designs D4, D5, and D6, for a 2⁴ factorial experiment. Designs D2 and D3 are orthogonal arrays of strength 2, D1 and D4 are balanced arrays of full strength, D5 is a balanced array of strength 2, and D6 is obtained from D3 by deleting the duplicate run. Designs D4 and D5 are also obtained by deleting a run from D1 and D2, respectively. Balanced arrays and orthogonal arrays are commonly used factorial designs in scientific experiments. “Search probabilities” are calculated for the comparison of search designs. Three criteria based on search probabilities are presented to determine the design which is most likely to identify the nonzero interaction. The calculation of these search probabilities depends on an unknown parameter ρ which has a signal-to-noise ratio form. For a given value of ρ, Criteria I and II are newly proposed in this paper and Criteria III is given in Shirakura et al. (Ann. Statist. 24 (6) (1996) 2560). We generalize Criteria I–III for all values of ρ so that the comparison of search designs can be made without requiring a specific value of ρ. We have developed simplified methods for comparing designs under these three criteria for all values of ρ. We demonstrate, under all three criteria, that the balanced array D1 is more likely to identify the nonzero interaction than the orthogonal arrays D2 and D3, and the design D4 is more likely to identify the nonzero interaction than the designs D5 and D6.The methods of comparing designs developed in this paper are applicable to other factorial experiments for searching one nonzero interaction of any order.     

Orthogonal arrays and ordered designs

In the first part of this paper, we give a short and direct construction of signed orthogonal array SOA(λ,t,k,v), for any set of parameters λ,t,k,v,t ⩽ k. We also construct a specific basis of the Z-module generated by any SOA(0,t,k,v), for any t,k,v,t ⩽ k. We will then construct an inite family of large set of disjoint ordered designs by applying G.B. Khosrovshahi and S. Ajoodani-Namini's method to Luc. Teirlinck's construction.     

Minimax versions of the two-stage <Emphasis Type="Italic">t</Emphasis> test

Let X be a N(μ, σ ²) distributed characteristic with unknown σ. We present the minimax version of the two-stage t test having minimal maximal average sample size among all two-stage t tests obeying the classical two-point-condition on the operation characteristic. We give several examples. Furthermore, the minimax version of the two-stage t test is compared with the corresponding two-stage Gauß test.     

Mixed covering arrays of strength three with few factors

Covering arrays with mixed alphabet sizes, or mixed covering arrays, are useful generalizations of covering arrays that are motivated by software and network testing. Suppose that there are k factors, and that the ith factor takes values or levels from a set G_i of size g_i. A run is an assignment of an admissible level to each factor. A mixed covering array, MCA(N;t,k,g₁g₂…g_k), is a collection of N runs such that for any t distinct factors, i₁,i₂,…,i_t, every t-tuple from G_i1×G_i2×?×G_it occurs in factors i₁,i₂,…,i_t in at least one of the N runs. When g=g₁=g₂=?=g_k, an MCA(N;t,k,g₁g₂…g_k) is a CA(N;t,k,g). The mixed covering array number, denoted by MCAN(t,k,g₁g₂…g_k), is the minimum N for which an MCA(N;t,k,g₁g₂…g_k) exists. In this paper, we focus on the constructions of mixed covering arrays of strength three. The numbers MCAN(3,k,g₁g₂…g_k) are determined for all cases with k∈{3,4} and for most cases with k∈{5,6}.     

Bounds on the number of constraints for balanced arrays of strength t

We describe a method, based on non-negative definiteness of moment matrices, for deriving upper bounds on the number of constraints in balanced arrays of strength t, involving two or more symbols. It is seen that the method covers, in particular, those due to Rafter and Seiden (1974) and Chopra (1982,1983).     

Quantile inference for heteroscedastic regression models

Consider the nonparametric heteroscedastic regression model Y=m(X)+σ(X)?, where m(·) is an unknown conditional mean function and σ(·) is an unknown conditional scale function. In this paper, the limit distribution of the quantile estimate for the scale function σ(X) is derived. Since the limit distribution depends on the unknown density of the errors, an empirical likelihood ratio statistic based on quantile estimator is proposed. This statistics is used to construct confidence intervals for the variance function. Under certain regularity conditions, it is shown that the quantile estimate of the scale function converges to a Brownian motion and the empirical likelihood ratio statistic converges to a chi-squared random variable. Simulation results demonstrate the superiority of the proposed method over the least squares procedure when the underlying errors have heavy tails.     

On the norm of alias matrices in balanced fractional 2m factorial designs of resolution 2l + 1

The norm $\text{6A6 = {tr(A′A)}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ of the alias matrix A of a design can be used as a measure for selecting a design. In this paper, an explicit expression for 6A6 will be given for a balanced fractional 2^m factorial design of resolution 2l + 1 which obtained from a simple array with parameters (m; λ₀, λ₁,…, λ_m). This array is identical with a balanced array of strength m, m constraints and index set {λ₀, λ₁,…, λ_m}. In the class of the designs of resolution V (l = 2) obtained from S-arrays, ones which minimize 6A6 will be presented for any fixed N assemblies satisfying (i) m = 4, 11 ? N ? 16, (ii) m = 5, 16 ? N ? 32, and (iii) m = 6, 22 ? N ? 40.     

Non Autonomous Semilinear Stochastic Evolution Equations

In this article, we first give a version with continuous paths for stochastic convolution ∫^t₀U(t, s)φ(s)dW(s) driven by a Wiener process W in a Hilbert space under weaker conditions. Based on the Picard approximation and the factorization method, we prove the existence, uniqueness and regularity of mild solutions for non-autonomous semilinear stochastic evolution equations with more general assumptions on the coefficients. As an application, we obtain the Feller property of the associated semigroup.